CBSE Class 10 Maths Revision Notes Chapter 6 Triangles 2026–27

Triangles Class 10 explains similarity, proportional sides, equal angles and key theorems used in geometry. For CBSE Class 10 Maths, this chapter covers similar figures, triangle similarity, area ratios and Pythagoras Theorem.

Triangles in Class 10 builds on the idea of congruence studied earlier. In this chapter, students learn about figures that have the same shape but may not have the same size.

Use these CBSE Class 10 Maths Revision Notes Chapter 6 to revise the chapter step by step. Start with similar figures, then move to similarity of triangles, Basic Proportionality Theorem, similarity criteria, area theorem and Pythagoras Theorem.

Key Takeaways

• Similar figures: Figures with the same shape but not necessarily the same size.
• Similar triangles: Triangles with equal corresponding angles and proportional corresponding sides.
• Basic Proportionality Theorem: A line parallel to one side of a triangle divides the other two sides in the same ratio.
• Pythagoras Theorem: In a right triangle, hypotenuse² = base² + perpendicular².

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CBSE Class 10 Maths Chapter 6 Triangles at a Glance

Class 10 Maths Chapter 6 Triangles focuses on similar triangles and their use in proving important geometry results. The chapter also connects proportional sides with area ratios and right triangles.

Similar Figures in Triangles Class 10

Similar figures have the same shape but may have different sizes. All congruent figures are similar, but similar figures need not be congruent.

Examples:

• All circles are similar.
• All squares are similar.
• All equilateral triangles are similar.
• A small photograph and its enlarged copy are similar.

Two polygons are similar when:

• their corresponding angles are equal
• their corresponding sides are in the same ratio

Copyable format:

Corresponding angles are equal.

Corresponding sides are proportional.

Similarity of Triangles

The similarity of triangles is based on equal corresponding angles and proportional corresponding sides. If both conditions are met, the triangles have the same shape.

For triangles ABC and DEF:

ΔABC ~ ΔDEF

This means:

∠A = ∠D

∠B = ∠E

∠C = ∠F

And:

AB/DE = BC/EF = AC/DF

Copyable format:

ΔABC ~ ΔDEF

AB/DE = BC/EF = AC/DF

The order of letters is important. In ΔABC ~ ΔDEF, A corresponds to D, B corresponds to E and C corresponds to F.

Basic Proportionality Theorem

The Basic Proportionality Theorem is also known as Thales theorem. It connects parallel lines with equal ratios in a triangle.

Statement:

If a line is drawn parallel to one side of a triangle and intersects the other two sides at distinct points, then the other two sides are divided in the same ratio.

In ΔABC, if DE || BC, where D lies on AB and E lies on AC, then:

AD/DB = AE/EC

Copyable format:

If DE || BC, then AD/DB = AE/EC.

This theorem is used when a parallel line is already given in the question.

Converse of Basic Proportionality Theorem

The converse of Basic Proportionality Theorem works in the reverse direction. It helps prove that a line is parallel to the third side of a triangle.

Statement:

If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

In ΔABC, if D lies on AB and E lies on AC, and:

AD/DB = AE/EC

Then:

DE || BC

Copyable format:

If AD/DB = AE/EC, then DE || BC.

Use this theorem when the ratio is given and parallel lines need to be proved.

Criteria for Similarity of Triangles

The criteria for similarity of triangles help students prove two triangles similar without checking all angles and all sides. The main criteria are AA, AAA, SSS and SAS.

AA and AAA Similarity Criteria

Two triangles are similar if two angles of one triangle are equal to two angles of another triangle. The third pair of angles becomes equal by the angle sum property.

If:

∠A = ∠D

∠B = ∠E

Then:

ΔABC ~ ΔDEF

Copyable format:

If ∠A = ∠D and ∠B = ∠E, then ΔABC ~ ΔDEF.

The AAA similarity criterion says that if all three corresponding angles are equal, the triangles are similar.

Copyable format:

If ∠A = ∠D, ∠B = ∠E and ∠C = ∠F, then ΔABC ~ ΔDEF.

SSS Similarity Criterion

The SSS similarity criterion is used when all corresponding sides of two triangles are in the same ratio. In this case, the corresponding angles are also equal.

If:

AB/DE = BC/EF = AC/DF

Then:

ΔABC ~ ΔDEF

Copyable format:

If AB/DE = BC/EF = AC/DF, then ΔABC ~ ΔDEF.

Use this criterion when the question gives all three sides or side ratios.

SAS Similarity Criterion

The SAS similarity criterion is used when two sides of one triangle are proportional to two sides of another triangle, and the included angles are equal.

If:

AB/DE = AC/DF

And:

∠A = ∠D

Then:

ΔABC ~ ΔDEF

Copyable format:

If AB/DE = AC/DF and ∠A = ∠D, then ΔABC ~ ΔDEF.

Use this criterion when two side ratios and the angle between them are given.

Areas of Similar Triangles

The areas of similar triangles are related to the squares of their corresponding sides. This theorem is important for questions that combine similarity and area.

Statement:

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

If:

ΔABC ~ ΔPQR

Then:

ar(ABC)/ar(PQR) = AB²/PQ² = BC²/QR² = AC²/PR²

Copyable format:

If ΔABC ~ ΔPQR, then

ar(ABC)/ar(PQR) = AB²/PQ² = BC²/QR² = AC²/PR²

This means side ratio is squared when comparing areas.

Pythagoras Theorem

Pythagoras Theorem applies to a right-angled triangle. It relates the hypotenuse, base and perpendicular.

Statement:

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Copyable format:

Hypotenuse² = Base² + Perpendicular²

For right ΔABC, right-angled at B:

AC² = AB² + BC²

Here:

• AC is the hypotenuse.
• AB and BC are the other two sides.

Converse of Pythagoras Theorem

The converse of Pythagoras Theorem helps check whether a triangle is right-angled. It works when the side lengths are known.

Statement:

If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Copyable format:

If AC² = AB² + BC², then ∠B = 90°.

Example:

For a triangle with sides 3 cm, 4 cm and 5 cm:

5² = 3² + 4²

25 = 9 + 16

25 = 25

So, the triangle is right-angled.

Formula and Theorem Table for Class 10 Triangles

This table can be used with Class 10 Mathematics Revision Notes Chapter 6 for quick theorem recall before solving geometry questions.

Solved Examples from Chapter 6 Triangles

Solved examples help students identify which theorem to use. First check whether the question gives parallel lines, angle equality, side ratios, area ratios or a right triangle.

Example 1: Use Basic Proportionality Theorem

In ΔABC, DE || BC. D lies on AB and E lies on AC.

Then, by Basic Proportionality Theorem:

AD/DB = AE/EC

If:

AD = 3 cm

DB = 2 cm

AE = 6 cm

Find EC.

Use:

AD/DB = AE/EC

3/2 = 6/EC

3EC = 12

EC = 4 cm

Answer:

EC = 4 cm

Example 2: Prove Two Triangles Similar by AA Criterion

In triangles ABC and DEF:

∠A = ∠D

∠B = ∠E

Since two angles of one triangle are equal to two angles of another triangle:

ΔABC ~ ΔDEF

Answer:

The triangles are similar by AA similarity criterion.

Example 3: Use Areas of Similar Triangles

If ΔABC ~ ΔPQR and:

AB/PQ = 3/2

Then:

ar(ABC)/ar(PQR) = AB²/PQ²

ar(ABC)/ar(PQR) = 3²/2²

ar(ABC)/ar(PQR) = 9/4

Answer:

The ratio of areas is 9:4.

Example 4: Use Pythagoras Theorem

In right ΔABC, right-angled at B:

AB = 6 cm

BC = 8 cm

Find AC.

Use:

AC² = AB² + BC²

AC² = 6² + 8²

AC² = 36 + 64

AC² = 100

AC = 10 cm

Answer:

AC = 10 cm

Chapter 6 Triangles Summary

These Class 10 Maths Chapter 6 Notes help students revise similar figures, triangle similarity, theorem ratios, area relation and Pythagoras Theorem in one flow.

• Similar figures have the same shape but not necessarily the same size.
• All congruent figures are similar, but all similar figures need not be congruent.
• Two triangles are similar when their corresponding angles are equal and corresponding sides are proportional.
• If DE || BC, then:
  • AD/DB = AE/EC
• If AD/DB = AE/EC, then:
  • DE || BC
• AA, AAA, SSS and SAS are used to prove similarity of triangles.
• For similar triangles:
  • ar(ABC)/ar(PQR) = AB²/PQ² = BC²/QR² = AC²/PR²
• In a right triangle:
  • Hypotenuse² = Base² + Perpendicular²
• The converse of Pythagoras Theorem helps identify a right-angled triangle.

How to Choose the Right Similarity Criterion

Students often lose marks because they know the criteria but choose the wrong one in a proof. Use the given information in the question to decide the criterion.

Quick rule:

• Use AA when angles are given.
• Use SSS when all three side ratios are given.
• Use SAS when two side ratios and the included angle are given.
• Use BPT when a parallel line is given.
• Use converse of BPT when a parallel line needs to be proved.

Proof Flow for Important Theorems in Triangles

Proof-based questions in Chapter 6 usually follow a fixed pattern. This table helps students revise how to start and complete each theorem proof.

Useful Important Questions Class 10 Maths Links